Lecture #7
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Transcript of Lecture #7
Lecture #7
What makes a good eye? Part I. Resolution
Land and Nilsson first half of chapter 3
2/14/13
Two key features of an eye
• Resolution (today)Precision with which eye can resolve
or distinguish a spatial image
• Sensitivity (next time)How much light can be detected by the
eye
Eye design
• Optical devices are subject to laws of physics and optimized by the engineers who design them
• Eyes are subject to laws of physics and optimized by evolution
Camera - transfers object image to “film”
Image resolution depends on pixel density
Vertebrate eye - single lens which transfers image to retina
Spatial resolution depends on cone spacing
Wolfe et al Sensory Perception fig 2.9
Human retinal mosaic in foveaJW temporal nasal AN nasal
Roorda and Williams 1999
Huge variation from person to person in distribution of cones and in M/L cone ratios!
Optimizing resolution
• Receptor size and spacing
• Lens focal length
• Optical imperfections
YouAreHere
v v
v
High Resolution Low Resolution
Why does this even exist?
Resolution - seeing one point of light
receptor
Reasons that points of light might end up as blurred points
of light
• Lens does not focus perfectlyAberration = imperfection
• Cornea, lens or humours scatter some of light
• DiffractionWhen light goes through an aperture,
it gets diffracted
Resolution
Optimal receptor spacing to resolve objects
• Need enough photoreceptors to define boundaries between objects
Optimal receptor spacing
• If receptors are too big (spaced too coarsely) they can’t resolve fine detail
Want optimal receptor spacing• No reason to have more receptors than this as
they won’t resolve the images any better
Refraction: Light bends at interface
More dense
Less dense
Bigger incident angle = more bending
Lens focal length, f
• Focal length is distance at which parallel light is focused
f
Lens focal length
• Determined by curvature of lens surface
Lens radius of curvature
Curvature as if cut lens out of surface of sphere with that radius
Lens focal length
• Determined by curvature of lens surface
Radius of curvature of sphere that matches lens’s curvature
Lens focal length is determined by curvature of lens surfaces
Large radius of curvature =
large focal length
Small radius of curvature =
small focal length
Calculate focal length from lens makers equation
Lens focal length
• Determined by curvature of lens surface
For thin lens
Lens focal length
If lens is symmetric, R1=-R2=R
If lens is flat on one side, R2=∞
How can we make focal length longer?
PhET lens simulator
Physics lens equation
Distance of image and object in relation to lens focal length
dO
Object Image
di
f
Physics lens equation
If dO, object distance is very large dO>>di
dO
O I
di
So di = f
Principal rays and nodal point
dO di
Note: both cornea and lens focus the light so retina is
located at combined focal length of lens and cornea
Retina
Eye resolution
Determine resolution based on how well can resolve a periodic pattern or grating
One period of grating
Quantifying resolution
• Minimum resolvable grating will match receptor spacing
• If receptor spacing is s then grating covers 2s on retina
• Angle = 2s / f
f
Angle
2s
One cycle of grating covers two photoreceptors
• Angle = 2s / f
• So can resolve one cycle if it covers no less than that angle
• Max resolvable spatial frequencyv = 1 / angle of one
grating cycle = f / 2s
s
f
Angle
Angles
retina
grating
2s
f
θTanθ= 2s / fBut for small θ tanθ=θSo θ= 2s/f
θ is in radians!
Interreceptor angle
Angle between two adjacent photoreceptors
θ = s / f
s
f
Angleθ
One cycle of grating covers two photoreceptors
• To cover full cycle of grating the angle is 2θ= 2s / f
• Maximum resolvable spatial frequency, vv = 1/2θ= f / 2s
s
f
Angle2θ
Minimum receptor acceptance angle
• Another angle:Δρ minimum
receptor acceptance angle
f eye focal length = distance to retina
d receptor diameter
Resolution =1/ Δρ = f/d
Δρ
f
d
In general s ≈ d and 2s/f ≈ 2d/f
Two ways to characterize eye resolution
• Maximum resolvable spatial frequency, v = f/2s
sf
Angle2θ=2s/f
Δρ
fd
Inter-receptor spacing, Δρ = d/f
Grating resolution = f / 2s Receptor resol = 1/Δρ= f/d = f /s
Some typical spatial frequencies - Vertebrate eyes
Animal Max resolvable spatial freq
Inter-receptor angle
Eagle 8000 cycles/rad
0.0036 deg
Human 4175 0.007
Cat 573 0.05
Goldfish 409 0.07
Rat 57 0.5
Some typical spatial frequencies - Vertebrate eyes
Animal Max resolvable spatial freq
Inter-receptor angle, Δρ
Eagle 8000 cycles/rad
0.0036 deg
Human 4175 0.007
Cat 573 0.05
Goldfish 409 0.07
Rat 57 0.5
Δρ= 0.0036º * π / 180 º = 6.28 x 10-5 rad
Some typical spatial frequencies - Vertebrate eyes
Animal Max resolvable spatial freq
Inter-receptor angle, Δρ
Eagle 8000 cycles/rad
0.0036 deg
Human 4175 0.007
Cat 573 0.05
Goldfish 409 0.07
Rat 57 0.5
Δρ= 0.0036º * π / 180 º = 6.28 x 10-5 rad
v = 1 / 2Δρ = 1 / (2*6.28 x 10-5) = 7960 cycles / rad
WHAT ABOUT ME?
How can we modify an eye to get highacuity night vision?
Chicken Owl
Image size relative to object size: Similar triangles
• f is focal length of the eye
• O = object size• I = image size
• U=distance to object
• O / U = I / f
f
Image size relative to object size: Similar triangles
• O / U = I / f
• f is focal length of the eye
• O = object size• I = image size
• U=distance to object
O
I
U
f
Magnification = I / O = f / U
PhET lens simulator
Spatial frequency = cycles per degree
Spatial frequency Eye resolution
Resolvable spatial frequency = f / 2s
• How can we make spatial frequency and therefore eye resolution increase?
Increasing eye resolution
• Spatial freq = resolution = f / 2s
• Decrease receptor size, sCan’t decrease below 2 μm Smaller receptors leak light
• Increase lens focal length, f
sf
Angle
What happens as f gets bigger?
• O / U = I / f
• O and U stay same
• What happens to I as f gets bigger?
f• If s stays the same, what happens?
Lens
• To increase resolution increase lens focal length, f
• Retina needs to be at a distance f from the focusing elements (lens / cornea combo)
• To increase f, what does this require?
Some typical spatial frequencies - Vertebrate eyes
Animal Max resolvable spatial freq
Inter-receptor angle
Eagle 8000 cycles/rad
0.0036 deg
Human 4175 0.007
Cat 573 0.05
Goldfish 409 0.07
Rat 57 0.5
Some typical spatial resolutions - Invert eyes
Animal Max resolvable spatial freq
Inter-receptor angle
Octopus 2632 cycles/rad
0.011 deg
Jumping spider 716 0.04
Worker bee 30 0.95
Crab 19 1.5
Drosophila 5.7 5
Nautilus 3.6 8
Flatworm 0.8 35
T
Rule of “thumb”
• If you view your thumb at arms length, it subtends about 2 degrees
• Human resolution is 4000 cycles / radSince π radians = 180 degreesThis is 73 cycles / deg
• So you can resolve 146 cycles across your thumb
Eye chart
Further down chart you can read, the better your visual acuity
Eye chart is based on sine gratings
1700 cycles / radian
Typical human spatial acuity
• Can resolve <0.01 deg
• Place letters at 20 ft
• Visual acuity = Resolution you have at 20
ft Distance at which
“normal” viewer would have same resolution as you
Limitations to eye resolution
• Diffraction• Optical aberrations
SphericalChromatic
Diffraction
• When light passes through an aperture or lens, it will be distorted by diffraction
Diffraction
• When light passes through a lens, it will be distorted by diffraction
Edges do not slow down as much as middleWhen arrive at retina, the edge and middle interfere
Diffraction
• Along center line will constructively interfere
• On sides, will destructively interfere
Diffraction
• Airy disk - width of interference peak is w = λ / D
D w
Calculate w two ways and compare
• λ = 650 nm (red laser) • D = 0.088 mm
Spot widthdistance
w= spot width / distance = 1 inch / 17 ft * 12 inch/ft
= 0.005 rad not too bad compared to expectation
Diffraction through aperture
• Distance from bright to first dark spot
Sin θ = λ / Dλ = wavelengthD = slit width
What does this mean for humans
• Pupil diameter = 2 mm• Visible light λ = 500 nm
Spatial frequency = 1/wSo humans can resolve 1/.00025
radians or about 4000 cycles / radian
Diffraction limit
Resolution of human cones in fovea
Cone diameter 2 μm2 s = 4 μmDistance 20 mm
Angle = 2s/f = 4 μm / 20mm
= 0.0002 rad
= 0.011°
Resolution of human cones in fovea
• Angular resolution based on cone spacing0.0002 rad or 0.011°
• Spatial frequency = 1/angular resolution5000 cycles / rad or 90 / deg
• Diffraction limit4000 cycles /radSo cone spacing is close to diffraction limit
Diffraction Spatial frequency = 1/w = D / λ
• How does an eagle see 8000 cycles / rad
• What has to change so it has better resolution (and overcomes diffraction?)
Imperfections in the visual system
• Diffraction when light goes through aperture
• Several other defects of lens systems
Fig 3.6
Optical defects 1. Lens focal defect
• Myopia
• Near sighted
• Cornea / lens focus too much
Optical defects 1. Lens focal defect
• Hyperopia
• Far sighted
• Cornea / lens focus not enough
2. Lens spherical aberration
• Light at edges is bent more than light in center of lens
2. Lens spherical aberration
• Light at edges is bent more than light in center of lens
3. Lens chromatic aberration
• Lens index of refraction and so focal length varies with wavelength • Light of different wavelengths are
focused to different points
Refraction as function of wavelength
λ n2 f
440 1.4155 12.03 cm486 1.410 12.19590 1.4015 12.45650 1.3983 12.55
Blue light bends moreWill focus in front of red light
Human lens index of refractionVaries between edge and center - correct for aberrations
Sivak and Mandelman 1982
Center inner outer periphery
Spherical aberration of human lens
1.386
1.406
Periphery focuses less than in center to exactly compensate for spherical aberration